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The idea in one paragraph

A language model does not give you an answer. It gives you a probability distribution over answers, and generating text draws one sample from it. Ask the same question twice and you can get two different results. Most people treat that as a defect.

We treat it as an untapped resource. Sample the same frozen model N times and select well, and accuracy climbs sharply — no training, no fine-tuning, no new weights. The knowledge was always in there. It just needed asking more than once.

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Why voting works so well

Here is the asymmetry that makes the whole thing run, and it is more elegant than it first looks:

Correct answers agree with each other. Wrong answers scatter.

A wrong trajectory has a thousand different ways to be wrong and picks a different one each time, so errors split into singletons. The correct answer is the only thing several attempts can converge on together.

That is why counting votes recovers answers only a small minority reached. The model does not need to be right most of the time — it needs to be right more consistently than it is wrong in any one particular way.

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Quickstart

from bestofn import BestOfN

engine = BestOfN("Qwen/Qwen2.5-0.5B-Instruct", n=16)
r = engine.solve("A train travels at 60 km/h for 3 hours. How far does it go?")

r.answer          # '180'
r.agreement       # 0.81   how strongly the trajectories agreed
r.effective_n     # 15     how many produced a usable answer

Works with any causal LM, at any N:

BestOfN("deepseek-ai/DeepSeek-R1-Distill-Qwen-1.5B", n=32)
BestOfN("meta-llama/Llama-3.1-8B-Instruct", n=16)
BestOfN("/path/to/your/local/model", n=128)

Both backends are tested on real hardware: transformers runs anywhere torch runs, and vllm is dramatically faster at large N — it is the one the measurements below were generated with.

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Measured results

Qwen2.5-0.5B-Instruct on GSM8K, 200 problems, 128 trajectories each, weights frozen. Every figure is recomputed from the published trajectories by scripts/analyse.py, which re-runs extraction over the raw reasoning text.

N	random	majority	95% CI	coverage
1	45.3%	45.3%	[40.3, 50.5]	45.3%
2	46.4%	46.5%	[37.0, 50.5]	56.7%
4	46.1%	53.0%	[45.0, 58.5]	66.6%
8	46.2%	58.3%	[52.0, 65.5]	75.0%
16	46.1%	61.8%	[55.0, 68.0]	81.7%
32	46.5%	64.0%	[56.0, 69.0]	86.7%
64	46.1%	65.5%	[58.5, 72.0]	90.6%
128	46.3%	66.5%	[60.5, 73.0]	93.5%

45.3% to 66.5% on a half-billion-parameter model, with the weights frozen throughout. Against random selection at the same N, exact McNemar gives p ≤ 6.6 × 10⁻⁵ at every one of 200 random seeds, with a median of 8.2 × 10⁻¹⁰.

We quote the worst seed rather than the best, and we say what it is: the worst of the 200 we enumerated, not a bound. random draws differently on every run, so its p-value has a distribution and a wider sweep will find a worse seed. The conclusion is what does not move — significant at all 200.

And coverage reaches 93.5%: on more than nine problems out of ten, this small model does find the right answer somewhere in its 128 attempts. That is the number that says how much is still on the table.

How much of the gain is really selection

Most reports skip this, and it is the part that decides whether a headline means anything:

N=1, a single sample	45.3%
N=128, random among the trajectories that answered	46.3%	+1.0
N=128, majority vote	66.5%	+20.2
		+21.2 total

Random selection improves slightly with N without selecting anything, because with more trajectories one of them usually did not abstain. Separating the two shows that 20.2 of the 21.2 points — 95% of the gain — is genuine selection, not an artefact of comparing a one-sample baseline against an N-sample system.

We report it this way because that comparison quietly folds the first number into the second, and the size of the fold is not knowable in advance. Here it is small. It is small because the token budget lets trajectories finish; at a tighter budget the same experiment would have credited five times as much of the gain to the method.

The selectors we can measure here, against the baseline

selector at N=128	accuracy	95% CI
random (exact expectation)	46.3%	[38.0, 52.0]
majority	66.5%	[60.5, 73.0]
self_certainty	66.5%	[60.5, 73.0]
oracle (diagnostic ceiling)	93.5%	[90.0, 96.5]

verifier and verifier_argmax are absent because the published trajectories carry no reward-model scores — this release ships no reward model, so there was nothing to score them with. Plug one in and the same table prints them.

The accounting

Trajectories generated	25,600
Cast a vote	24,788 — 96.8%
Abstained	812 — 3.2%
Truncated at the token limit	216 — 0.8% (215 of them abstained)
Tokens generated	8,434,157
Re-extraction drift on replay	9

All 25,600 trajectories are published in full.

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It tells you what to do next

Beyond raising accuracy, Best-of-N measures the two halves of the problem separately and tells you which one you are actually facing. The framing is not ours — pass@k beside maj@n appears in the evaluation harnesses and in Snell et al. (2024). What is ours is that it takes one call, and that the answer arrives as a decision rather than as two numbers to interpret:

r = engine.solve(problem)

r.is_correct(gold)   # did the system return the right answer?
r.covered(gold)      # did any trajectory find it at all?

returned	reachable	What it means	What to do
✓	✓	Working	You are done. Consider whether you need this much N
✗	✓	The answer is in there	A selection problem — a verifier recovers it
✗	✗	Not yet reachable	Raise N, improve the prompt, or use a stronger model

Two numbers, one decision. Without them you are guessing whether to spend your next hour on sampling or on selection.

The library also reports the accounting most tooling hides:

r.effective_n     # trajectories that actually voted
r.n_abstained     # produced no usable answer
r.n_truncated     # ran out of tokens
r.total_tokens    # what it cost

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The same pool always gives the same answer

Shuffle the trajectory pool and every selector returns the same answer. random is the exception, and only because picking at random is what it is for.

That sounds like it should be free. It is not, and it took three separate things to make true:

The equivalence classes cannot be built greedily. Symbolic equivalence is not transitive — a parser accepts 0.3333333333 against 1/3, and 1/3 against 0.33333333333333, while rejecting the two decimals against each other. Compare each new answer against one representative per class and the partition you get depends on which answer the model emitted first. We take the transitive closure over all pairs instead.

Weights cannot be accumulated in arrival order. Float addition is not associative, so two permutations of the same weights can differ in the last bit — enough to flip a near-tie in a verifier-weighted vote. Totals are summed with math.fsum over a sorted list, which is exactly rounded.

Ties cannot break on whichever came first. Every tie, in every selector, resolves on the canonical key.

We check it in the test suite, not by hand. tests/test_selectors.py fuzzes all five deterministic selectors over pools built to hit the cases that break invariance — exact ties, duplicate scores, weights spread across many orders of magnitude, and pools past the merge cap — and separately asserts that the weight tally itself is unchanged by reordering. Each of the four historical bugs has been confirmed to turn that suite red when reintroduced.

It is visible in the published output too: at N=128 the resampled curve draws 128 trajectories from a pool of 128, so every draw is a permutation of one pool and the reported spread is 0.00. That is a necessary condition rather than a proof, which is why the tests carry the weight.

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The habit worth building: always print the baseline

r = engine.solve(problem, n=16)

r.select_with("random", seed=0)     # the honest baseline
r.select_with("majority")           # how much better is it, really?

Re-running a selector over an existing result is free — generation is the expensive part — so there is no reason not to. Our own published tables carry the random row, because a gain is only a gain relative to something.

Method	Needs	What it is for
random	nothing	The baseline. Print it every time
majority	nothing	The default, and it is strong
self_certainty	logprobs=True	Weighs votes by the model's own confidence
verifier	a verifier callable	The one that can promote a minority answer
verifier_argmax	a verifier callable	Single best trajectory, no vote
oracle	the gold answer	Measures your headroom during development

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Does it hold on other models?

That is the first question anyone asks after a single 0.5B result, so we ran the same protocol across a size range — same 200 GSM8K problems, same prompt, same temperature, same token budget, same N, weights frozen throughout. A comparison where the protocol drifts between rows is not a comparison.

model	one sample	Best-of-N	gain	coverage
HuggingFaceTB/SmolLM2-1.7B-Instruct 1.7B, different family · N=64	21.7%	59.0%	+37.3	86.5%
Qwen/Qwen2.5-0.5B-Instruct 0.5B general · N=64	45.5%	65.5%	+20.0	92.5%
Qwen/Qwen2.5-1.5B-Instruct 1.5B general · N=64	68.8%	80.0%	+11.2	97.0%
microsoft/Phi-3-mini-4k-instruct 3.8B, different family · N=64	80.9%	91.0%	+10.1	100.0%
Qwen/Qwen2.5-3B-Instruct 3B general · N=64	83.0%	91.0%	+8.0	98.5%
Qwen/Qwen2.5-Math-1.5B-Instruct 1.5B maths-tuned · N=64	84.3%	89.5%	+5.2	95.0%
Qwen/Qwen2.5-7B-Instruct 7B general · N=64	90.1%	94.0%	+3.9	98.5%
Qwen/Qwen3.8-27B-FP8 27B vision-language, text-only, FP8 · N=64	95.1%	97.5%	+2.4	98.5%

Every one of the 8 models we measured improved, by between +2.4 and +37.3 points, and none of them was trained. The gain shrinks as the base model gets better — which is what should happen, and is worth saying plainly rather than hiding behind the largest number in the table.

The row that matters more is coverage. It stays above what the vote returns on every model, by a median of 9.0 points: even the strongest one here is still failing to return answers it already found. That gap is the whole reason to work on selection rather than on sampling harder.

Every row is re-derived from that model's own published trajectories, with each answer re-extracted from the raw reasoning text rather than read back from storage — so a change to the extractor moves these numbers, and we would see it. Run python scripts/run_models.py --rescore results/models and they come out again on a CPU in under a minute.

Bring your own reward model

Best-of-N works with any published reward model, and it makes them work correctly:

from bestofn import BestOfN
from bestofn.verifiers import from_hub

verifier = from_hub("openbmb/Eurus-RM-7b")          # Apache-2.0
engine = BestOfN("your/model", n=16, verifier=verifier)
engine.solve(problem, method="verifier")

Reward models emit unbounded logits, not probabilities, and a naive implementation silently degrades to a plain majority vote when you hand it one — leaving you convinced your verifier is running when it is not. This one catches it, tells you exactly what to do, and the adapters apply the sigmoid for you.

Licences differ and they govern how you may use the scores. from_hub flags anything non-permissive, and verifiers.license_of(model_id) checks the Hub live. The full table is in USAGE.md.

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Everything here is checkable

pip install "bestofn[math]"
python scripts/analyse.py     # no GPU needed

The [math] extra pulls in math-verify, which is what makes 1/2, 0.5 and \frac{1}{2} count as one answer. Without it the script still runs and still reports, but equivalent answers written differently vote separately and the numbers come out slightly lower. Regenerating the figures additionally needs matplotlib; the analysis itself does not.

The published dataset contains the complete reasoning text of every trajectory, with finish_reason, log-probabilities and token counts — not answers extracted earlier by someone else. The analysis re-runs extraction over that raw text, computes exact McNemar against the random baseline, and reports bootstrap confidence intervals.

That means the numbers above are not asserted, they are reproduced — by you, in about two minutes on a laptop CPU, with no hardware. Very little published work in this area can say that.

To regenerate from scratch:

python scripts/run_gsm8k.py --backend vllm --n 128 --batch 25

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Where it shines

Best-of-N pays off most on:

• Tasks with one comparable final answer — mathematics, multiple choice, short factual questions, unit-testable code.
• Models that are sometimes right. The gain is largest when per-sample accuracy sits around 20–60%, which is exactly where small models live on hard problems.
• Hardware you already own. On your own GPU the extra samples cost you nothing but time you were not using. This is the one setting where the economics are simply free.

Practical guidance on choosing N, plugging in a verifier and measuring your own task is in USAGE.md.

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How this compares

Checked against the current published wheels in August 2026, not against reputation. Every cell below names the file it came from, so you can verify it yourself rather than take our word for it.

	lm-eval 0.4.12	dspy 3.3.1	bestofn
Majority vote	2 task configs, GSM8K only	majority()	yes
Answer comparison	one regex	lowercase + strip punctuation	symbolic (math-verify)
Coverage in the same table	no	no	yes
Random baseline	no	no	yes
Abstention accounted for	no	None is skipped	yes, published
Same answer whatever the order	no	no	yes
Replay re-extracts from raw text	writes samples, re-counts them	no	yes

Where the numbers came from. lm-eval's self-consistency is lm_eval/tasks/gsm8k/gsm8k-cot-self-consistency.yaml and its gsm8k_platinum twin — two files, offering score-first, maj@8 and maj@64. Its own comment on the maj@8 filter reads "Using a better estimator would be optimal": it takes the first 8 of the 64 rather than resampling, which is biased. Extraction is the single pattern The answer is (\-?[0-9\.\,]*[0-9]+), so a correct answer written any other way scores as wrong, and pass@k lives in an unrelated task (lm_eval/tasks/cruxeval/utils.py) rather than beside maj@n.

dspy.BestOfN (dspy/predict/best_of_n.py) is a different thing that shares the name: it takes a reward_fn, returns the single highest-scoring completion, and stops early at a threshold. It does not vote. dspy.majority (dspy/predict/aggregation.py) does, normalising with normalize_text — lowercase and strip punctuation — and breaking ties towards the earlier completion, so the answer depends on the order the samples arrive in.

trl 1.12.0 has no BestOfNSampler. It was removed; there are zero matches for it in the current wheel.

What we are not claiming. The idea is not ours. Sampling a model many times and taking the mode is self-consistency (Wang et al., 2022); measuring what the pool reached against what the selector returned is the subject of Brown et al.'s Large Language Monkeys (2024) and of Snell et al. (2024). Those papers did this before us and at greater scale. What was missing was a library that does it in one call and reports the parts honestly by default — and a published dataset you can re-derive every figure from without a GPU.

Where we lose. lm-eval covers hundreds of tasks; we measure one benchmark across eight models. We ship no reward model and no process reward model, so the 27-point gap between what our models reach and what they return is a gap we name rather than close. There is no beam search, no lookahead, no compute-optimal allocation — the selectors here are majority voting and a confidence weighting, and on our data the confidence weighting does not beat the free one. If you need throughput, vLLM and SGLang generate the trajectories and this only chooses among them.

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Built on solid ground

Inference-time compute is one of the most active areas in the field, and this sits squarely inside it:

• Cobbe et al., Training Verifiers to Solve Math Word Problems, 2021
• Wang et al., Self-Consistency Improves Chain of Thought Reasoning, 2022
• Lightman et al., Let's Verify Step by Step, 2023
• Snell et al., Scaling LLM Test-Time Compute Optimally, 2024
• Brown et al., Large Language Monkeys, 2024

What this adds: the selection layer the serving stacks deliberately leave out — vLLM removed best_of in 2025, SGLang discourages n>1, LMDeploy supports only 1 — a single small API over six interchangeable selectors, and the raw trajectories behind every number we publish.

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What happened to 1.0

Version 1.0 is withdrawn. Its numbers should not be cited, including by us.

It reported 83.3% coverage on AIME 2024 and credited a trained outcome reward model with a 16.6-point gain over majority voting. An adversarial audit of the published package found defects in answer extraction and in the handling of truncated trajectories that corrupted the inputs to the vote:

• Fractions and radicals collapsed onto their first digit, so one half, one third and minus one half all compared equal.
• Commas were deleted unconditionally, so the ordered pair (3,4) became the integer 34 — a fabricated answer, not a lost one.
• A trajectory cut off mid-reasoning contributed "the last number in its text" as a vote, indistinguishable from a real one.

Those measurements are not reproducible with the corrected implementation. Separately, the reward model was never published, the problem set it was measured on was never published, and in the one evidence file that was, it selected the same answer as majority voting on all 30 problems — a net gain of zero.

1.1 ships no reward model. It provides adapters for third-party ones, with the range validation that stops a reward model's unbounded logits from silently degrading a weighted vote into an unweighted one.

Everything in this card is GSM8K, measured with the version in the badge above, and reproducible from the trajectories published alongside it. The full account is in the technical note and the changelog.

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Citation

@software{arecesrivera2026bestofn,
  title  = {Best-of-N: inference-time compute for language models},
  author = {Areces Rivera, Alejandro},
  year   = {2026},
  doi    = {10.5281/zenodo.21936832},
  url    = {https://github.com/voidlinestudios12-jpg/Interlace-AI}
}

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License

Apache License 2.0 — free to use, modify and redistribute, including commercially.

Copyright 2026 Alejandro Areces Rivera — Interlace AI

Questions and collaboration: interlaceIA@gmail.com Release notes: CHANGELOG.md